Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.0.701 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&20\\12&9\end{bmatrix}$, $\begin{bmatrix}15&12\\8&15\end{bmatrix}$, $\begin{bmatrix}27&48\\46&1\end{bmatrix}$, $\begin{bmatrix}37&36\\42&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.q.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $32256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^8\cdot5^4\cdot7^4}\cdot\frac{(12x-y)^{48}(66082400676385652736x^{16}+250021294356993933312x^{15}y+310338740637747118080x^{14}y^{2}-39791843012530667520x^{13}y^{3}+57217567248450846720x^{12}y^{4}-64438276211777470464x^{11}y^{5}+5083758578550177792x^{10}y^{6}+4362161433642270720x^{9}y^{7}-1140924023959265280x^{8}y^{8}-424099028270776320x^{7}y^{9}+48052501996326912x^{6}y^{10}+59216201401319424x^{5}y^{11}+5112008450277120x^{4}y^{12}+345638229995520x^{3}y^{13}+262077208908480x^{2}y^{14}-20527487874432xy^{15}+527485056301y^{16})^{3}}{(12x-y)^{48}(72x^{2}+7y^{2})^{4}(72x^{2}-72xy-7y^{2})^{8}(72x^{2}+28xy-7y^{2})^{8}(72x^{2}+168xy-77y^{2})^{2}(792x^{2}+168xy-7y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.e.1.14 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.h.2.20 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.h.2.25 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.l.1.16 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.l.1.18 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.